Remark

Tag 08B3, permanent. Cite it as [Bourbaki, Tag 08B3].

Functions of a Real Variable, chapter III, § 2. EXPANSIONS OF THE EXPONENTIAL AND CIRCULAR FUNCTIONS AND OF THE FUNCTIONS ASSOCIATED WITH THEM

Since the remainder in formula (1) is $> 0$ for $x > 0$ one has, for $x > 0$

e^x > 1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^{n+1}}{(n+1)!}

and a fortiori

e^x > \frac{x^{n+1}}{(n+1)!}

for every integer $n$ : one deduces from this that $e^x / x^n$ tends to $+\infty$ with $x$ , for every integer $n$ : we shall find this result again in chap. V by another method (V, p. 231).

$^{2}$ CH. HERMITE proved in 1873 that $e$ is a transcendental number over the field $\mathbf{Q}$ of rational numbers (in other words, it is not the root of any polynomial with rational coefficients) (Œuvres, t. III, p. 150, Paris (Gauthier-Villars), 1912).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

V, p. 231